Method and device for synchronizing the carrier frequency of an offset quadrature phase-modulated signal

ABSTRACT

A method and a device for synchronizing the carrier frequency of a carrier signal comprising a frequency offset and/or a phase offset. According to the invention, the method estimates the frequency offset and/or phase offset of the carrier signal by means of a maximum likelihood estimation from a received signal, which comprises temporally discrete, complex rotary indices, for which only the temporally discrete phases are dependent on the frequency offset and/or phase offset. An offset quadrature-modulated received signal is thus converted into a modified received signal comprising temporally discrete, complex rotary indices, for which only the temporally discrete phases are dependent on the frequency offset and/or the phase offset.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a method and a device for carrier-frequencysynchronization in the case of an offset quadrature-phase-modulatedsignal.

2. Related Technology

When transmitters and receivers are synchronized with one another in atransmission system, the clock-pulse and carrier signal are matchedrespectively with regard to phase position and frequency at thetransmitter-end and receiver-end. The carrier-frequency synchronizationto be considered below presupposes a received signal synchronized withregard to the clock-pulse signal.

DE 103 09 262 A1 describes a method for carrier-frequencysynchronization of a signal with digital symbol sequences, in which thefrequency and/or phase offset of the carrier signal is estimated fromthe demodulated received signal by means of maximum-likelihoodestimation. The received signal containing the digital symbol sequenceconsists of complex rotary phasors associated with the individualsampling timing points, of which the discrete phases are dependent onlyupon the required frequency and/or phase offset of the carrier signal,and of which the discrete amplitudes are dependent only upon the digitalsymbol values of the demodulated received signal. The maximum-likelihoodestimation of the required frequency and/or phase offset of the carriersignal is implemented by maximization of the likelihood function, whichis formed from the sum of the real components of all time-discrete,complex rotary phasors of the received signal. The maximization of thelikelihood function is implemented by rotating the complex rotary phasorof the received signal associated with each sampling timing point in theclockwise direction at the level of the required frequency and/or phaseoffset so that it is disposed on the real axis. In this manner, it ispossible to obtain the required frequency and/or phase offset of thecarrier signal by observing the extreme values of the likelihoodfunction in each case separately for the frequency and/or phase offset.

In the context of DE 103 09 262 A1, the time-discrete received signalprovides one complex rotary phasor at each sampling timing point, ofwhich the phase value is dependent only upon the frequency and/or phaseoffset of the carrier signal, and of which the amplitude value isdependent upon the symbol value of the received signal sequence to betransmitted at the respective sampling timing point. A time-discretereceived signal of this kind is based upon a comparatively simplemodulation, for example, a conventional binary-phase phase sampling(BPSK). By contrast, if a more complex modulation method, especially anoffset quadrature phase modulation (offset-QPSK-modulation) is used,considerable inter-symbol interference, which additionally disturbs thephases of the time-discrete complex rotary phasors, occurs in thereceived signal because of the squaring and also because of the phasedisplacement of the in-phase relative to the quadrature component at thelevel of half of the symbol period in the context of an offset QPSKmodulation. Accordingly, the phases of the time-discrete complex rotaryphasors of the received signal are not only dependent upon the frequencyand/or phase offset of the carrier signal. In this case, the use of amaximum-likelihood estimation for the estimation of the requiredfrequency and/or phase offset of the carrier signal in the sense of themethod and the device known from the DE 103 09 262 A1 therefore fails toachieve the object.

SUMMARY OF THE INVENTION

The invention accordingly provides a method and a device for theestimation of the frequency and/or phase offset in the carrier signal inthe case of an offset quadrature-phase-modulated received signal using amaximum-likelihood estimation.

The invention provides a method for carrier-frequency synchronizationand by a device for carrier-frequency synchronization.

Accordingly, the invention provides a method for carrier-frequencysynchronization of a carrier signal influenced by a frequency and/orphase offset by means of estimating the frequency and/or phase offset ofthe carrier signal by estimating maximum-likelihood from a receivedsignal with time-discrete complex rotary phasors, in which only thetime-discrete phases are dependent upon the frequency and/or phaseoffset, wherein the received signal is an offset quadrature-modulatedreceived signal, which is converted for the maximum-likelihoodestimation via a pre-filtering step into a modified received signal withtime-discrete complex rotary phasors, in which only the time-discretephases are dependent upon the frequency and/or phase offset, and ofwhich the real components are maximized for the maximum-likelihoodestimation of the frequency and/or phase offset.

-   -   The invention also provides a device for carrier-frequency        synchronization of a carrier signal influenced by a frequency        and/or phase offset with a maximum-likelihood estimator for the        estimation of the frequency and/or phase offset of the carrier        signal from a received signal with time-discrete complex rotary        phasors, in which only the time-discrete phases are dependent        upon the frequency and/or phase offset, wherein the        maximum-likelihood estimator is preceded by a pre-filter and a        signal-processor, which converts the received signal, which is        an offset-quadrature-phase-modulated received signal, into a        modified received signal with time-discrete complex rotary        phasors, in which only the time-discrete phases are dependent        upon the frequency and/or phase offset, and wherein the        maximum-likelihood estimator maximizes the real components of        the complex rotary phasors for the estimation of the frequency        and/or phase offset.

According to the invention, the offset quadrature-phase-modulatedreceived signal is converted, after a demodulation, a sampling with anover-sampling factor of typically eight and a pre-filtering with asignal-matched pre-filter, in three successive signal-processing stages,into a modified received signal, of which the time-discrete complexrotary phasors each provide phases, which depend exclusively upon thefrequency and/or phase offset of the carrier signal used.

The first signal-processing stage is a further sampling, which generatesa time-discrete received signal with two sampling values per symbolperiod. This newly-sampled received signal therefore contains in eachdiscrete complex rotary phasor an additional phase dependent upon therespective sampling timing point, which is compensated in the subsequentsecond signal-processing stage by a complex multiplication with acomplex rotary phasor with a phase inverse to the latter. In a thirdsignal-processing stage, the received signal, with the additional phaseremoved from the respective time-discrete complex rotary phasor, isfinally subjected to a modulus-scaled squaring in order to ensure thatthe amplitude of each time-discrete complex rotary phasor of theaccordingly-modified received signal provides a positive value.

Accordingly, with the method according to the invention and the deviceaccording to the invention for carrier-frequency synchronization, amodified received signal, of which the time-discrete complex rotaryphasors each provide phases, which are dependent exclusively upon thefrequency and/or phase offset of the carrier signal used, is formed fromthe offset quadrature-phase-modulated received signal.

The time-discrete phases of the modified time-discrete received signalare then determined via an argument function, and a phase characteristicis formed from these. This phase characteristic of the modified receivedsignal, which is periodic over the period 2·π and discontinuous, is then“stabilized” at the discontinuity points to form a continuous phasecharacteristic of the modified received signal.

A continuous phase characteristic of a modified, offsetquadrature-phase-modulated received signal generated in this manner canbe expediently subjected to a maximum-likelihood estimation in the senseof DE 103 09 262 A1, in order to determine any frequency and/or phaseoffset occurring in the carrier signal used for a subsequentcarrier-frequency synchronization of the received signal.

BRIEF DESCRIPTION OF THE DRAWINGS

A preferred embodiment of the method and the device according to theinvention for carrier-frequency synchronization are explained in greaterdetail below with reference to the drawings. The drawings are asfollows:

FIG. 1 shows an expanded block circuit diagram of the transmissionsystem;

FIG. 2 shows a reduced block circuit diagram of the transmission system;

FIG. 3 shows a block circuit diagram of the device according to theinvention for carrier-frequency synchronization;

FIG. 4 shows a complex phasor diagram of a received signal modifiedaccording to the invention;

FIG. 5 shows a time characteristic for a discontinuous phasecharacteristic and for the “stabilized” continuous phase characteristicof a received signal modified according to the invention; and

FIG. 6 shows a flow chart for the method according to the invention forcarrier-frequency synchronization.

DETAILED DESCRIPTION

Before describing an embodiment of the method according to the inventionand of the device according to the invention for carrier-frequencysynchronization in the case of an offset QPSK signal in greater detailwith reference to FIGS. 3 to 6, the following section of the descriptionprovides a derivation of the mathematical basis required in thiscontext.

The starting point is a complex baseband model of a transmission system1 for continuous-time complex signals, for which the expanded blockcircuit diagram is presented in FIG. 1.

The complex symbol sequence s(t) of an offset QPSK signal to betransmitted as shown in equation (1) is supplied to the input 2 of thetransmission system 1:

$\begin{matrix}{{s(t)} = {{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {\delta\left( {t - {n\; T_{S}}} \right)}}} + {j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {\delta\left( {t - {n\; T_{S}}} \right)}}}}}} & (1)\end{matrix}$

In this context a_(R)(n) and a_(l)(n) represent the symbol values forthe in-phase and quadrature components of the offset QPSK transmissionsignal to be generated, which can adopt, for example, the real values{±s_(i)} of the symbol alphabet. The symbol sequences of the in-phaseand quadrature components are periodic in each case with regard to thesymbol length T_(s). In terms of system theory, the symbol sequence s(t)to be transmitted is convoluted in the transmission filter 3 with theassociated impulse response h_(s)(t) and supplies the filtered symbolsequence s_(F)(t) according to equation (2) at the output of thetransmission filter 3:

$\begin{matrix}{{s_{F}(t)} = {{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{S}\left( {t - {n\; T_{S}}} \right)}}} + {j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{S}\left( {t - {n\; T_{S}}} \right)}}}}}} & (2)\end{matrix}$

The subsequent lag element 4 models the time offset ε·T_(S) occurring asa result of absent or inadequate synchronization between the transmitterand the receiver, which is derived from the timing offset ε. In thiscontext, the timing offset ε can adopt positive and negative valuestypically between ±0.5. The filtered symbol sequence s_(ε)(t) taking thetiming offset ε·T_(S) into consideration at the output of the lagelement 4 is therefore obtained according to equation (3):

$\begin{matrix}\begin{matrix}{{s_{ɛ}(t)} = {{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{S}\left( {t - {ɛ\; T_{S}} - {n\; T_{S}}} \right)}}} +}} \\{j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{S}\left( {t - {ɛ\; T_{S}} - {n\; T_{S}}} \right)}}}}\end{matrix} & (3)\end{matrix}$

The lag-influenced, filtered symbol sequence s_(ε)(t) is mixed in anoffset QPSK modulator—modelled in FIG. 1 as a multiplier 5—with acomplex carrier signal e^(j(2π(f) ^(T) ^(+Δf)t+Δφ)) to form an offsetQPSK-modulated transmission signal s_(HF)(t). The carrier signale^(j(2π(f) ^(T) ^(+Δf)t+Δφ)) has a carrier frequency f_(T), whichprovides a frequency offset Δf and a phase offset Δφ as a result of theabsence of carrier-frequency synchronization. Ignoring signal errors ofthe quadrature modulator, such as overdrive in the carrier signal on thein-phase or respectively quadrature channel, gain imbalance between thein-phase and quadrature channel, quadrature error between the in-phaseand the quadrature channel, the mathematical relationship of the offsetQPSK-modulated transmission signal s_(HF)(t) is obtained as shown inequation (4):

$\begin{matrix}\begin{matrix}{{s_{HF}(t)} = {\begin{bmatrix}{{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {{h_{S}\left( {t - {ɛ\; T_{S}} - {n\; T_{S}}} \right)}++}}}\mspace{50mu}} \\{j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{S}\left( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {n\; T_{S}}} \right)}}}}\end{bmatrix} \cdot}} \\{{\mathbb{e}}^{j{({{2\;{\pi{({f_{T} + {\Delta\; f}})}}\; t} + {\Delta\;\varphi}})}}}\end{matrix} & (4)\end{matrix}$

By comparison with the symbol sequence of the quadrature component ofthe offset QPSK-modulated transmission signal s_(HF)(t), the symbolsequence of the interface component is phase-displaced by one symbolperiod.

On the transmission path between the transmitter and the receiver, anadditive white Gaussian noise (AWGN) n(t) is superposed additively onthe offset QPSK-modulated transmission signal s_(HF)(t), which providesa real and imaginary component n_(R)(t) and n_(l)(t) as shown inequation (5)n(t)=n _(R)(t)+j·n _(l)(t)  (5)

The received signal r_(HF)(t) received by the receiver is thereforeobtained according to equation (6):r _(HF)(t)=s _(HF)(t)+n(t)  (6)

In the receiver, the offset QPSK-modulated received signal r_(HF)(t)with superposed noise n(t) is mixed down into the baseband with thecarrier signal e^(−j2πf) ^(T′) in a demodulator modelled in FIG. 1 asthe multiplier 6. The demodulated received signal r(t) at the output ofthe demodulator 6, which contains an in-phase and quadrature symbolsequence distorted with the frequency and phase offset of the carriersignal, is obtained according to equation (7):

$\begin{matrix}\begin{matrix}{{r(t)} = {{{{s_{ɛ}(t)} \cdot {\mathbb{e}}^{j\;{({{2\;{\pi\Delta}\; f} + {\Delta\;\varphi}})}}} + {n(t)}} =}} \\{= {\begin{bmatrix}{{{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{S}\left( {t - {ɛ\; T_{S}} - {n\; T_{S}}} \right)}}} +}\mspace{65mu}} \\{j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{S}\left( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {n\; T_{S}}} \right)}}}}\end{bmatrix} \cdot}} \\{{\mathbb{e}}^{j{({{2\;\pi\;\Delta\; f\; t} + {\Delta\;\varphi}})}} + {n(t)}}\end{matrix} & (7)\end{matrix}$

As shown in equation (7), the system-theoretical effects of themodulator 5 and of the demodulator 6 of the transmission system 1 on theoffset QPSK-modulated signal are partially cancelled, so that themodulator 5 and the demodulator 6 in FIG. 1 can be replaced by a singlemultiplier 7 as shown in the reduced block circuit diagram of FIG. 2,which mixes the lag-influenced, filtered symbol sequence s_(E)(t) with asignal e^(j(2πΔft+Δφ)) according to equation (8) to provide atransmission signal s_(NF)(t) in the baseband.

$\begin{matrix}\begin{matrix}{{s_{NF}(t)} = \begin{bmatrix}{{{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{S}\left( {t - {ɛ\; T_{S}} - {n\; T_{S}}} \right)}}} +}\mspace{65mu}} \\{j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{S}\left( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {n\; T_{S}}} \right)}}}}\end{bmatrix}} \\{{\mathbb{e}}^{j{({{2\;\pi\;\Delta\; f\; t} + {\Delta\;\varphi}})}}}\end{matrix} & (8)\end{matrix}$

The transmission signal s(t) with superposed additive white Gaussiannoise n(t) as shown in the reduced block circuit diagram in FIG. 2 isreceived in the receiver as a received signal r(t), which corresponds tothe received signal according to equation (7) of the extended blockcircuit diagram shown in FIG. 1.

The received signal r(t) is convoluted in the receiver filter 8 as shownin equation (9) with the associated impulse response h_(E)(t) and leadsto the signal e(t) at the output of the receiver filter 8, whichrepresents an in-phase and quadrature symbol sequence filtered anddistorted with reference to signal error and frequency and phase offset:e(t)=r(t)*h _(E)(t)  (9)

The received signal r(t) is filtered by the receiver filter 8 with theimpulse response h_(E)(t) and provides the filtered received signal e(t)at its output. The receiver filter 8 is a signal-matched filter. Asignal-matched filter according to equation (10) provides an impulseresponse h_(E)(t) corresponding to the impulse response h_(s)(t) of thetransmitter filter 3 and accordingly a transmission function H_(E)(f)reflected relative to the transmission function H_(s)(f) as shown inequation (11):h _(E)(t)=h _(S)(−t)  (10)H _(E)(f)=H _(S)(−f)  (11)

In this manner, the signal-noise distance of the filtered receivedsignal e(t) is maximized as a ratio of the useful power relative to theinterference power.

Following the receiver filter 8, a sampling of the filtered receivedsignal is implemented in a sampling and holding element 9 referred tobelow as the second sampling and holding element with a sampling ratef_(A), which is increased by comparison with the symbol frequency f_(s)of the received signal r(t) by the over-sampling factor os. In thiscontext, the over-sampling factor os provides a value of 8, as shown indetail in [1]: K. Schmidt: “Digital clock-pulse recovery for band-widthefficient mobile telephone systems”, 1994, ISBN 3-18-14 75 10-6.

After the sampling of the filtered received signal e(t), anotherpre-filtering of the signal is implemented in a pre-filter 10. Thepurpose of the pre-filter 10 is to minimize the data-dependent jitter inthe signal. For this purpose, the frequency spectrum H_(E)(f) of thereceiver filter 8 is linked multiplicatively to the frequency spectrumH_(V)(f) of the pre-filter 10 according to equation (12) to form acombined frequency spectrum H_(EV)(f):H _(EV)(f)=H _(E)(f)·H _(V)(f)  (12)

If the transmitter filter 3 according to equation (13) provides afrequency spectrum H_(s)(t), which corresponds to a root-cosine filterwith a roll-off factor r, the common frequency spectrum H_(EV)(f) of thereceiver filter 8 and of the pre-filter 10 according to equation (14)must be designed dependent upon the frequency spectrum H_(s)(f) of thetransmitter filter 3 in order to minimize data-dependent jitter in thereceived signal r(t) as shown in [1].

$\begin{matrix}{{H_{S}(f)} = \left\{ \begin{matrix}1 & {{{for}\mspace{14mu}{f}} < \frac{f_{S}}{2}} \\{\cos\left\lbrack {\frac{\pi{f}}{2\; r\; f_{S}} - \frac{\pi\;\left( {1 - r} \right)}{4\; r}} \right\rbrack} & {{{for}\mspace{14mu}\left( {1 - r} \right)\frac{f_{S}}{2}} < {f} \leq {\left( {1 + r} \right)\;\frac{f_{S}}{2}}} \\0 & {{{for}\mspace{14mu}\left( {1 + r} \right)\frac{f_{S}}{2}} < {f}}\end{matrix} \right.} & (13) \\{{H_{EV}(f)} = \left\{ \begin{matrix}{{H_{S}\left( {f - f_{S}} \right)} + {H_{S}\left( {f + f_{S}} \right)}} & {{{for}\mspace{14mu}{f}} \leq {\frac{f_{S}}{2}\left( {1 + r} \right)}} \\{random} & {{{for}\mspace{14mu}\frac{f_{S}}{2}\left( {1 + r} \right)} < {f} \leq f_{S}} \\0 & {{{for}\mspace{14mu} f_{S}} < {f}}\end{matrix} \right.} & (14)\end{matrix}$

According to equation (15), the frequency response H_(GES)(f) can beinterpreted as a low-pass filter H_(GES0)(f) symmetrical to thefrequency f=0 with a bandwidth of

${\frac{f_{S}}{2} \cdot r},$which is frequency-displaced in each case by

${\pm \frac{f_{S}}{2}}\text{:}$

$\begin{matrix}\begin{matrix}{{H_{GES}(f)} = {{{H_{{GES}\; 0}(f)}*\left( {{\delta\left( {f - \frac{f_{S}}{2}} \right)} + {\delta\left( {f + \frac{f_{S}}{2}} \right)}} \right)} =}} \\{= {{H_{{GES}\; 0}\left( {f - \frac{f_{S}}{2}} \right)} + {H_{{GES}\; 0}\left( {f + \frac{f_{S}}{2}} \right)}}}\end{matrix} & (15)\end{matrix}$

The corresponding impulse response h_(GES)(t) is therefore obtained asshown in equation (16):

$\begin{matrix}{{h_{GES}(t)} = {{{h_{{GES}\; 0}(t)} \cdot \left( {{\mathbb{e}}^{j\; 2\;\pi\;\frac{f_{s}}{2}t} + {\mathbb{e}}^{{- j}\; 2\;\pi\;\frac{f_{s}}{2}t}} \right)} = {{h_{{GES}\; 0}(t)} \cdot {\cos\left( {2\;\pi\;\frac{f_{S}}{2}t} \right)}}}} & (16)\end{matrix}$

The signal v(t) at the output of the pre-filter 10 can be obtainedaccordingly by replacing the impulse response h_(s)(t) of thetransmitter filter 3 in the baseband according to equation (8) with theimpulse response h_(GES)(t) of the overall transmission system in thetransmitter signal s_(NF)(t) as shown in equation (17):

$\begin{matrix}{{v(t)} = {{{s_{NF}(t)}*{h_{GES}(t)}} = {\left\lbrack {{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{GES}\left( {t - {ɛ\; T_{S}} - {n\; T_{S}}} \right)}}} + {j\; \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{GES}\left( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {n\; T_{S}}} \right)}}}}} \right\rbrack \cdot {\mathbb{e}}^{j\;{({{2\;\pi\;\Delta\;{ft}} + {\Delta\;\varphi}})}}}}} & (17)\end{matrix}$

Starting from equation (16), the impulse responseh_(GES)(t−εT_(S)−nT_(S)) can be described according to equation (18)

$\begin{matrix}{{h_{GES}\left( {t - {ɛ\; T_{S}} - {n\; T_{S}}} \right)} = {{h_{{GES}\; 0}\left( {t - {ɛ\; T_{S}} - {nT}_{S}} \right)} \cdot \left( {- 1} \right)^{n} \cdot {\cos\left( {2\;\pi\;\frac{f_{S}}{2}\left( {t - {ɛ\; T_{S}}} \right)} \right)}}} & (18)\end{matrix}$

Similarly, the mathematical relationship of equation (19) can bedetermined for the impulse response

${h_{GES}\left( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {n\; T_{S}}} \right)}.$

$\begin{matrix}{{h_{GES}\left( {t - \;{ɛ\; T_{S}} - \frac{T_{S}}{2} - {n\; T_{S}}} \right)} = {{{h_{{GES}\; 0}\left( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {n\; T_{S}}} \right)} \cdot \left( {- 1} \right)^{n} \cdot \sin}\;\left( {2\;\pi\;\frac{f_{S}}{2}\left( {t - {ɛ\; T_{S}}} \right)} \right)}} & (19)\end{matrix}$

On the basis of the mathematical terms in equations (118) and (19), thecombinations shown in equation (20) and (21), and therefore themathematical context for the output signal v(t) of the pre-filter 10 inthe event of an excitation of the transmission system 1 with an offsetQPSK signal s(t), can be transferred from equation (17) to equation(22).

$\begin{matrix}{{R(t)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{{GES}\; 0}\left( {t - {ɛ\; T_{S}} - {n\; T_{S}}} \right)} \cdot \left( {- 1} \right)^{n}}}} & (20) \\{{I(t)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{{GES}\; 0}\left( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {n\; T_{S}}} \right)} \cdot \left( {- 1} \right)^{n}}}} & (21) \\{{v(t)} = {\left\lbrack {{{R(t)} \cdot {\cos\left( {2\;\pi\;\frac{f_{S}}{2}\left( {t - \;{ɛ\; T_{S}}} \right)} \right)}} + {{j\; \cdot {I(t)} \cdot \sin}\;\left( {2\;\pi\;\frac{f_{S}}{2}\left( {t - \;{ɛ\; T_{S}}} \right)} \right)}} \right\rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{ft}} + {\Delta\;\varphi}})}}}} & (22)\end{matrix}$

The signal v(t) at the output of the pre-filter 10 according to equation(22) is delayed in a downstream delay element 11 by the timing offset−{circumflex over (ε)}·T_(S). The estimated timing offset {circumflexover (ε)}, which is determined by an estimation unit not illustratedhere for the estimation of the timing offset {circumflex over (ε)} of anoffset QPSK-modulated signal, corresponds, with optimum clock-pulsesynchronization, to the actual timing offset E of the offsetQPSK-modulated signal v(t). In this case, timing offset is removedcompletely from the output signal v_(ε)(t) of the delay element 11according to equation (23).

$\begin{matrix}{{v_{ɛ}(t)} = {\left\lbrack {{{R_{ɛ}(t)} \cdot {\cos\left( {2\;\pi\;{\frac{f_{S}}{2} \cdot t}} \right)}} + {{j\; \cdot {I_{ɛ}(t)} \cdot \sin}\;\left( {2\;\pi\;{\frac{f_{S}}{2} \cdot t}} \right)}} \right\rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t}} + {\Delta\;\phi}})}}}} & (23)\end{matrix}$

The associated combinations R_(ε)(t) and I_(ε)(t) with the timing offsetε·T_(S) removed are obtained according to equations (24) and (25):

$\begin{matrix}{{R_{ɛ}(t)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{{GES}\; 0}\left( {t - {n\; T_{S}}} \right)} \cdot \left( {- 1} \right)^{n}}}} & (24) \\{{I_{ɛ}(t)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{{GES}\; 0}\left( {t - \frac{T_{S}}{2} - {n\; T_{S}}} \right)} \cdot \left( {- 1} \right)^{n}}}} & (25)\end{matrix}$

Equations (23), (24) and (25) show that the time-synchronized receivedsignal v_(ε)(t) is not in the time-discrete form required for a use ofthe maximum-likelihood method for determining the estimatedfrequency-offset and phase-offset value Δ{circumflex over (f)} andΔ{circumflex over (φ)} according to equation (26):r(t′)=|r(t′)|·e ^(j(2πΔft′+Δφ))  (26)

The time-synchronized received signal v_(ε)(t) is therefore convertedaccording to the invention into a form corresponding to equation (26) aswill be shown below.

For this purpose, if the output signal v_(ε)(t) of the delay unit 11 isobserved only at the discrete timing points

${t^{\prime} = {{\mu \cdot \frac{T_{S}}{2}}\mspace{14mu}\left( {{\mu = 0},1,2,{{\ldots\mspace{11mu}{2 \cdot N}} - 1}} \right)}},$the output signal v_(ε)(t′) of the delay unit 11 is composed, accordingto equations (27a), (27b), (27c) and (27d) and dependent upon the timingpoint observed, only of a purely real or purely imaginary component anda complex rotary phasor e^(j(2πΔf·t′+Δφ)):

$\begin{matrix}{{t^{\prime} = {{{0 \cdot \frac{T_{S}}{2}}\text{:}\mspace{14mu}{v_{ɛ}\left( t^{\prime} \right)}} = {\left\lbrack {R_{ɛ}\left( t^{\prime} \right)} \right\rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}}{{v_{ɛ}(t)} = {{R_{ɛ}(t)} \cdot {\mathbb{e}}^{j\;{({0 \cdot \frac{\pi}{2}})}} \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t}} + {\Delta\;\phi}})}}}}} & \left( {27a} \right) \\{{t^{\prime} = {{{1 \cdot \frac{T_{S}}{2}}\text{:}\mspace{14mu}{v_{ɛ}\left( t^{\prime} \right)}} = {\left\lbrack {j \cdot {I_{ɛ}\left( t^{\prime} \right)}} \right\rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}}{{v_{ɛ}\left( t^{\prime} \right)} = {{I_{ɛ}\left( t^{\prime} \right)} \cdot {\mathbb{e}}^{j\;{({1\frac{\pi}{2}})}} \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}} & \left( {27b} \right) \\{{t^{\prime} = {{{2 \cdot \frac{T_{S}}{2}}\text{:}\mspace{14mu}{v_{ɛ}\left( t^{\prime} \right)}} = {\left\lbrack {- {R_{ɛ}\left( t^{\prime} \right)}} \right\rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}}{{v_{ɛ}\left( t^{\prime} \right)} = {{R_{ɛ}\left( t^{\prime} \right)} \cdot {\mathbb{e}}^{j\;{({2 \cdot \frac{\pi}{2}})}} \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}} & \left( {27c} \right) \\{{t^{\prime} = {{{3 \cdot \frac{T_{S}}{2}}\text{:}\mspace{14mu}{v_{ɛ}\left( t^{\prime} \right)}} = {\left\lbrack {{- j} \cdot {I_{ɛ}\left( t^{\prime} \right)}} \right\rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}}{{v_{ɛ}\left( t^{\prime} \right)} = {{I_{ɛ}\left( t^{\prime} \right)} \cdot {\mathbb{e}}^{j\;{({3\frac{\pi}{2}})}} \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}} & \left( {27d} \right)\end{matrix}$

According to equations (24) and (25), the combinations R_(ε)(t′) andI_(ε)(t′) represent the real-value low-pass signals, which can be eitherpositive or negative because of the statistical distribution of thesymbol sequences a_(r)(n) and a_(l)(n). In the paragraphs below, theyare each described by the time-dependent, real-value amplitude A(t′).Accordingly, instead of individual timing-point-related equations (27a),(27b), (27c) and (27d) for the output signal v_(ε)(t′) of the delayelement 11, a single mathematical equation (28) containing all of thetiming points is obtained at the individual timing points

${t^{\prime} = {{\mu \cdot \frac{T_{S}}{2}}\left( {{\mu = 0},1,2,{{\ldots\mspace{14mu}{2 \cdot N}} - 1}} \right)}}\mspace{14mu}$for the output signal v_(ε)(t′) of the delay element 11:

$\begin{matrix}{{v_{ɛ}\left( t^{\prime} \right)} = {{{{A\left( t^{\prime} \right)} \cdot {\mathbb{e}}^{j{({\mu \cdot \frac{\pi}{2\;}})}} \cdot {\mathbb{e}}^{j{({{2\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}\mspace{14mu}{for}\mspace{14mu} t^{\prime}} = {\mu \cdot \frac{T_{S}}{2}}}} & (28)\end{matrix}$

If the time-discrete output signal v_(ε)(t′) of the delay element 11 atthe individual sampling times

${t^{\prime} = {{{\mu \cdot \frac{T_{S}}{2}}\mu} = 0}},1,2,{{\ldots\mspace{14mu}{2 \cdot N}} - 1}$is phase-displaced by a timing-point-related phase angle

${{- \mu} \cdot \frac{T_{S}}{2}},$a phase-displaced, time-discrete signal w(t′), from which, by comparisonwith the time-discrete output signal v_(ε)(t′) of the delay element 11,the complex term

${\mathbb{e}}^{{j\mu}\frac{T_{S}}{2}}$has been removed, is obtained according to equation (29) from thetime-discrete output signal v_(ε)(t′) of the delay element 11:w(t′)=A(t′)·e ^(j(2πΔf·t′+Δφ))  (29)

Since the amplitude A(t′) of the signal w(t′) can adopt positive andnegative values, a modulus-formation of the amplitude A (I′) should beimplemented. A modulus formation of the amplitude of a complex signal isimplemented by squaring with subsequent division by the modulus. Thephase of the complex signal is doubled by this process, but the modulusremains unchanged.

The application of squaring and subsequent division by the modulus tothe signal w(t′) leads to the signal x(t′) according to equation (30),which can be interpreted as a time-discrete complex rotary phasor with atime-discrete amplitude |A (t′)| and a time-discrete phase2·(2πΔft′+Δφ)=2·(φωμT_(S)+Δφ) as shown in FIG. 5, and which is in theform according to equation (26) appropriate for a maximum-likelihoodestimation of the frequency and phase offset of the carrier signal:x(t′)=|A(t′)|·e ^(j2(2πΔf−t′+Δφ)) +n(t′)  (30)

Moreover, equation (30) takes into consideration the additiveinterference n(t′), which, in a good approximation, is uncorrelated andprovides a Gaussian distribution. Accordingly, the optimum estimatedvalue for Δf und Δφ is obtained by the maximization of themaximum-likelihood function L(Δ{circumflex over (f)},Δ{circumflex over(φ)}), which, according to equation (31) corresponds to a maximizationof the real components of all time-discrete, complex rotary phasors ofthe signal x(t′), and an estimated value Δ{circumflex over (f)} andΔ{circumflex over (φ)} for the frequency and phase offset of the carriersignal can be determined.

$\begin{matrix}{{L\left( {{\Delta\;\overset{\bullet}{f}},{\Delta\;\overset{\bullet}{\varphi}}} \right)} = {{Re}\left\{ {\sum\limits_{\mu}\;{{x\left( {t^{\prime} = {\mu\frac{T_{S}}{2}}} \right)} \cdot {\mathbb{e}}^{- {{j2}{({{2{\pi\Delta}\overset{\bullet}{f}\mu\frac{T_{S}}{2}} + {\Delta\overset{\bullet}{\varphi}}})}}}}} \right\}}} & (31)\end{matrix}$

The maximization of the real components of all time discrete complexrotary phasors of the signal x(t′) can be interpreted as a “turningback” of the time -discrete, complex rotary phasor of the signal x(t′)in each case by the phase angle

${2 \cdot \left( {{2\pi\;\Delta\; f\;\mu\frac{T_{S}}{2}} + {\Delta\;\varphi}} \right)},$until these coincide with the real axis in the complex plane.

Starting from this derivation of the mathematical basis, the followingparagraphs describe the device according to the invention forcarrier-frequency synchronization in the case of an offsetQPSK-modulated signal as shown in FIG. 3 and of the method according tothe invention for carrier-frequency synchronization in the case of anoffset QPSK-modulated signal as shown in FIG. 6.

In the device according to the invention shown in FIG. 3, theclock-pulse synchronized output signal v_(ε)(t) of the delay element isunder-sampled in a sampling and holding element 12 referred to below asthe first sampling and holding element at two sampling values per symbolperiod T_(s).

The output signal v_(ε)(t′) of the first sampling and holding element 12is supplied to a complex multiplier 13, in which it is subjected to asampling timing-point-related phase displacement through the phase angle

${- \mu} \cdot {\frac{T_{S}}{2}.}$

The output signal w(t′) of the complex multiplier 13 accordinglyphase-displaced relative to the signal v_(ε)(t′) is supplied to a unitfor modulus-scaled squaring 14, consisting of a squarer followed by adivision by the modulus, in which the modulus of its amplitude is formedand its phase is doubled.

The signal at the output of the unit for modulus-scaled squaring 14represents the modified received signal x(t′), which thesignal-processor 15 has generated from the clock-pulse-synchronizedreceived signal v_(ε)(t) by under-sampling in the first sampling andholding element 12, by phase-displacement in the complex multiplier 13and by forming the modulus of the amplitude and doubling the respectivephase in the unit for modulus-scaled squaring 14.

In a subsequent maximum-likelihood estimator 18, the estimated valuesΔ{circumflex over (f)} and Δ{circumflex over (φ)} for the frequency andphase offset of the carrier signal are determined from thetime-discrete, modified received signal x(t′), as described, forexample, in DE 103 09 262 A1.

A frequency-offset and phase-offset estimator, which avoids 2·πslips—so-called “cycle slips”—occurring in the phase characteristic,which result through small amplitudes of the time-discrete, modifiedreceived signal x(t′) from the superposed interference in the case of aphase regression, as described, for example, in DE 103 09 262 A1, can beused as a maximum-likelihood estimator. The phase regression cannottherefore be used for this application.

The method according to the invention for carrier-frequencysynchronization of an offset QPSK-modulated signal is described belowwith reference to FIG. 6.

As shown in FIG. 6, procedural stage S10 of the method according to theinvention for carrier-frequency synchronization of an offsetQPSK-modulated signal provides a demodulation of the received signalr(t) according to equation (7). Through an appropriate design of thereceiver to transmitter filter, a signal-matched filtering of thereceived signal r(t), which leads to an optimization of the signal-noisedistance in the received signal r(t), is implemented at the same time asthe demodulation.

In the next procedural stage S20, in a second sampling unit, thereceived signal r(t) is sampled with an over-sampling factor typicallywith a value of 8.

The sampled received signal is supplied in procedural stage S30 to apre-filter according to equation (17), which minimizes data-dependentjitter in the received signal r(t).

A time-synchronization of the sampled, filtered and modulated receivedsignal v(t) is provided in the next procedural stage S40 according toequation (23) by means of a delay element, which obtains the estimatedtiming offset {circumflex over (ε)} from an estimator, which is notdescribed in greater detail here.

In the next procedural stage S50, an additional sampling—a firstsampling—of the time-synchronized received signal v_(ε)(t) isimplemented at two sampling values per symbol period T_(s) as shown inequation (28).

A sampling timing-point-related phase displacement of theadditionally-sampled, time-synchronized received signal v_(ε)(t′) isimplemented by complex multiplication with a samplingtiming-point-related multiplication factor

${\mathbb{e}}^{{- {j\mu}}\frac{T_{S}}{2}}$in order to compensate the respective inverse, complex factor

${\mathbb{e}}^{{j\mu}\frac{T_{S}}{2}}$in the received signal v_(ε)(t′) according to equation (29) in the nextprocedural stage S60.

The next procedural stage S70 provides the modulus formation of thetime-discrete amplitudes A(t′) and squaring of the time-discrete phases2πΔft′+Δφ of the phase-displaced, additionally-sampled andtime-synchronized received signal w(t′) according to equation (30).

In the next procedural stage S80, the time-discrete, modified receiversignal x(t′) obtained from the time-synchronized received signalv_(ε)(t) in procedural stages S50, S60 and S70 by means of asignal-processor 15 is used to determine its time-discrete, continuousphase characteristic α′(t′).

In procedural stage S80, the estimated frequency-offset and phase-offsetvalues Δ{circumflex over (f)} and Δ{circumflex over (φ)} of the carriersignal are determined according to equation (31) by means ofmaximum-likelihood estimation. The modified received signal x(t′) isused for this purpose. The maximum-likelihood estimator used in thiscontext should ideally be able to deal with phase slips—so-called “cycleslips”—resulting from interference signals superposed on the modifiedreceived signal v_(ε)(t) at small amplitudes of the modified receivedsignal v_(ε)(t), and is disclosed, for example, in DE 103 09 262 A1.

1. Method for carrier-frequency synchronization of a carrier signalinfluenced by a frequency and/or phase offset comprising estimating thefrequency and/or phase offset of the carrier signal by estimatingmaximum-likelihood from a received signal with time-discrete complexrotary phasors, wherein only the time-discrete phases are dependent uponthe frequency and/or phase offset, and wherein the received signal is anoffset quadrature-modulated received signal, converting the receivedsignal for the maximum-likelihood estimation via a first pre-filteringstep into a modified received signal with time-discrete complex rotaryphasors wherein the conversion of the received signal comprises a firstsampling at two sampling values per symbol period, a complexmultiplication, and a modulus-scaled squaring, the modulus-scaledsquaring comprising implementing squaring and inverse modulus formation,and maximizing the real components of the complex rotary phasors for themaximum-likelihood estimation of the frequency and/or phase offset,wherein the conversion of the received signal is preceded by ademodulation, a second sampling, a second pre-filtering, and aclock-pulse synchronization.
 2. Method for carrier-frequencysynchronization according to claim 1, comprising implementing thecomplex multiplication with a complex phase angle e^(−jμπ/2), wherein μis a sampling index.
 3. Method for carrier-frequency synchronizationaccording to claim 1, comprising following the conversion of the offsetquadrature-modulated received signal with the maximum-likelihoodestimation of the frequency and phase offset of the carrier signal. 4.Device for carrier-frequency synchronization of a carrier signalinfluenced by a frequency and/or phase offset comprising: amaximum-likelihood estimator for the estimation of the frequency and/orphase offset of the carrier signal from a received signal withtime-discrete complex rotary phasors, in which only the time-discretephases are dependent upon the frequency and/or phase offset, and apre-filter and a signal-processor preceding the maximum-likelihoodestimator, wherein the pre-filter and signal processor convert thereceived signal, wherein the received signal is anoffset-quadrature-phase-modulated received signal, and is modified withtime-discrete complex rotary phasors, the device further comprising oneor more of the group consisting of: (a) a demodulator and a delayelement for clock-pulse synchronization connected upstream of thesignal-processor, (b) a second sampling unit connected upstream of thepre-filter, and (c) the maximum-likelihood estimator connecteddownstream of the signal-processor, wherein the maximum-likelihoodestimator maximizes the real components of the complex rotary phasorsfor the estimation of the frequency and/or phase offset, and wherein thesignal-processor comprises a first sampling unit, a complex multiplier,and a unit for modulus-scaled squaring, and wherein the unit formodulus-scaled squaring comprises a squarer, a modulus former connectedin parallel to the squarer, and a divider connected downstream of thesquarer and the modulus former.
 5. Device for carrier-frequencysynchronization of claim 4, comprising the demodulator and the delayelement for clock-pulse synchronization connected upstream of thesignal-processor.
 6. Device for carrier-frequency synchronizationaccording to claim 4, comprising the second sampling unit connectedupstream of the pre-filter.
 7. Device for carrier-frequencysynchronization according to claim 4, comprising the maximum-likelihoodestimator connected downstream of the signal-processor.
 8. Method forcarrier-frequency synchronization of a carrier signal influenced by afrequency and/or phase offset comprising: estimating the frequencyand/or phase offset of the carrier signal by estimatingmaximum-likelihood from a received signal with time-discrete complexrotary phasors, wherein only the time-discrete phases are dependent uponthe frequency and/or phase offset, and wherein the received signal is anoffset quadrature-modulated received signal; converting the receivedsignal for the maximum-likelihood estimation via a first pre-filteringstep into a modified received signal with time-discrete complex rotaryphasors, wherein the conversion of the received signal comprises a firstsampling at two sampling values per symbol period, a complexmultiplication, and a modulus-scaled squaring, the modulus-scaledsquaring comprising implementing squaring and inverse modulus formation;following the conversion of the offset quadrature-modulated receivedsignal with the maximum-likelihood estimation of the frequency and phaseoffset of the carrier signal; and maximizing the real components of thecomplex rotary phasors for the maximum-likelihood estimation of thefrequency and/or phase offset.
 9. Method for carrier-frequencysynchronization according to claim 8, comprising implementing thecomplex multiplication with a complex phase angle e^(−jμπ/2), wherein μis a sampling index.